| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 7-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s + 6·13-s + 14-s − 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s − 21-s + 4·22-s − 24-s − 25-s + 6·26-s − 27-s + 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.261191561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.261191561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.45153405687664, −15.00764210685304, −14.13628135697101, −14.00465663803868, −13.33450710105923, −13.00628371150186, −12.23267727478700, −11.58789344356589, −11.32395245887194, −10.78383038793804, −10.11751421013583, −9.354315112498741, −9.086460685820588, −8.161498223162854, −7.569253591723007, −6.605178876037725, −6.383642334921889, −5.813395405074304, −5.283244446734589, −4.500894003428410, −3.892934852573126, −3.306732262096598, −2.236628178172096, −1.508666306115535, −0.9578244148465305,
0.9578244148465305, 1.508666306115535, 2.236628178172096, 3.306732262096598, 3.892934852573126, 4.500894003428410, 5.283244446734589, 5.813395405074304, 6.383642334921889, 6.605178876037725, 7.569253591723007, 8.161498223162854, 9.086460685820588, 9.354315112498741, 10.11751421013583, 10.78383038793804, 11.32395245887194, 11.58789344356589, 12.23267727478700, 13.00628371150186, 13.33450710105923, 14.00465663803868, 14.13628135697101, 15.00764210685304, 15.45153405687664
